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Let a connected $d$-regular expander graph have $n$ vertices, edge-expansion constant $h>0$, and $d\ge 2$. For $n\ge 16d^2$, its least bi-Lipschitz distortion into L1 space satisfies the explicit boundNormalize an embedding so that each image distance is at least the graph distance and at most $D$ times it. Every image edge has length at most $D$. The L1 Poincare inequality for an expander graph then gives $D\ge (h/d)\overline{d}$, where $\overline{d}$ is the mean ordered-pair graph distance.

A ball of radius $r$ has at most $1+d+\cdots +d^r\le 2d^r$ vertices. Take $r=\lfloor \log (n/4)/\log d\rfloor$. Its size is at most $n/2$, so at least half the vertices are farther than $r$ from each fixed vertex. Hence $\overline{d}\ge r/2$. The size assumption implies $r\ge (\log n)/(2\log d)$, proving the bound. For a fixed-degree family with expansion bounded below, distortion therefore grows at least proportionally to $\log n$.

Choose $D=d^2$ and a fixed $d$-regular graph $H$ on $D^2$ vertices with absolute nonconstant eigenvalues at most $1/10$. Start with the loop-inclusive complete graph $G_0$ on $D$ vertices and iterate $G_{t+1}=G_t^2\mathrm{zigzag}H$. The degree stays $D$, vertex counts are $D^{2t+1}$, and ${\rho }_{t+1}\le {\rho }_t^2+21/100$. Since ${\rho }_0=0$, all ${\rho }_t\le 1/2$. The spectral cut bound gives multiplicity-counted edge expansion at least $D/4$. Removing loops and merging parallel edges loses at most a factor $D$, producing simple expander graphs with maximum degree $D$ and edge expansion at least $1/4$.